Yes, she will be able to get on and off the cruise ship without a passport. I believe the only Caribbean cruise port that requires a passport at this time is Martinique.
Passports are still recommended. If for any reason your friend misses the cruise ship at any port and/or has to fly back to the states due to an emergency, she will need a passport. If she does not have one she can be delayed for days or even weeks in that foreign country and the little bit of money she thought she'd save by not getting a passport will cost her a whole lot more.
US Passport cost = $145 and it is good for 10 years, that comes to less than $15/year to have one. It's a small price to pay for the peace of mind it will give her when leaving the country. She has plenty of time to get one, be sure she knows the downsides to not having one.